A Girsanov type representation of quadratic-exponential cost functionals for linear quantum stochastic systems

TL;DR

This paper introduces a Girsanov type representation for quadratic-exponential cost functionals in linear quantum stochastic systems, enabling analysis of quantum risk-sensitive performance criteria through a novel quantum Karhunen-Loeve expansion.

Contribution

It develops a finite-horizon quantum Karhunen-Loeve expansion and a Girsanov type representation for QEFs, applicable regardless of the system-field state.

Findings

Derived a quantum Karhunen-Loeve expansion for system variables.

Established a Girsanov type representation for quadratic-exponential functions.

Connected QEFs to the moment-generating functional of system variables.

Abstract

This paper is concerned with multimode open quantum harmonic oscillators and quadratic-exponential functionals (QEFs) as quantum risk-sensitive performance criteria. Such systems are described by linear quantum stochastic differential equations driven by multichannel bosonic fields. We develop a finite-horizon expansion for the system variables using the eigenbasis of their two-point commutator kernel with noncommuting position-momentum pairs as coefficients. This quantum Karhunen-Loeve expansion is used in order to obtain a Girsanov type representation for the quadratic-exponential functions of the system variables. This representation is valid regardless of a particular system-field state and employs the averaging over an auxiliary classical Gaussian random process whose covariance operator is defined in terms of the quantum commutator kernel. We use this representation in order to relate the QEF to the moment-generating functional of the system variables. This result is also specified for the invariant multipoint Gaussian quantum state when the oscillator is driven by vacuum fields.